Based on our current partial understanding of the laws of physics, it does appear that, to the extent to which the results of physical experiments are governed by mathematical theories that we understand, they are computable, and in fact computable on a quantum computer manipulating at most $10^{122}$ bits. This means that all occurrences of real numbers in the theory are prevented from introducing incomputability.
Let me sketch how this is supposed to go. An amplitude of a particular outcome in a particular experiment might be expressed as a path integral, i.e. an integral over the infinite-dimensional space of paths a particle can take. To compute this, we could try to argue that this integral is well-approximated by an integral over a finite-dimensional space of piecewise linear paths, as long as the steps are sufficiently small. We would then argue that the integral over a finite-dimensional space could be approximated by the sum over a large number of random points in the space. By approximating the function we're integrating, we approximate the amplitude. (Of course it must be possible for physicists to computably approximate the predictions of their theories, so that they can test them in experiments. Methods like Feynman diagrams and lattice QFT enable them to do this in practice.)
Now the key point is that, whatever experiment we perform only allows us to approximate the amplitude. If we perform an experiment $n$ times, we can estimate the probabilities, and thus the amplitudes, to within an error of $~1/\sqrt{n}$. So performing the experiment $2^k$ times we only learn $k$ bits of the amplitude. Moreover, because the cosmological constant is positive, only a bounded amount of matter can fit in our observable universe, containing a bounded amount of entropy, with which we can only perform the experiment a bounded number of times before the heat death of the universe.
This phenomenon is also what shields the values of physical constants from our understandings. Only a bounded number of bits of the fine structure constant need to be known to predict every experiment that occurs in our universe.
Beyond the difficulties presented by the actual rules of the actual universe, there are a couple philosophical difficulties with your idea. The first is that, if such a theory were true, we would have difficulty testing it, as its predictions would be independent of ZFC. So what you would require is a physical phenomenon whose behavior is described by an elegant matheamtical model, which for some experiments produces predictions that can be calculated within ZFC, and for other experiments produces predictions that are independent of ZFC. Furthermore, we would have to be highly confident that no other, more mathematically sedate, theory, predicts the same behavior on the first set of experiments by a different mathematical formalism - e.g. by using whatever mathematical tools we were using to guess the answer.
I think sufficiently strong evidence for a theory of this type would indeed convince people of what you want, although there would still be dissenting views - for instance that the universe is a simulation and that the answers reflect only what the creators of the simulation think the answers to these ZFC-independent problems are, rather than the true answers. There is just no evidence I am aware of for any such theories.
